Project B14

Project B14 - Stability of Micro- and Macroscopic Traffic Models on the Transition from Circular Road to Infinite Lanes

Background and Motivation

Mathematical models of traffic flow have been succesfully used to describe, understand and predict traffic phenomena like jams, behaviour at bottlenecks, etc.
Traditionally, models for single-lane vehicular traffic are often formulated either "microscopically" as systems of ODEs, trying to capture the dynamics at the level of a single car, or "macroscopically" as (systems of) PDEs, describing e.g. car density and flow velocity. For microscopic models describing a finite number of cars on a ring road, detailed stability and bifurcation analysis can be done. They explain why and how a slight variation of circumstances like mean density, reaction time, or driving behaviour can lead to an abrupt change
of the behaviour of smoothly flowing traffic.
The fact that the information about a driver's decisions influences his behaviour at a later point of time by travelling upstream might lead to unrealistic effects. If, however, an open road of infinite length is considered instead, the situation becomes more complex.

Aims and Objectives

The aim of this project is to study how stability properties of traffic flow models change on the transition from circular road to infinite lane and from microscopic to macroscopic description. The notions of convective, remnant, and absolute stability, well-known from PDEs, are to be introduced for microscopic models by considering their behaviour under certain exponentially weighted norms.
Analysis of the Bando model on the circular road has shown that periodic solutions corresponding to stop-and-go-waves emerge from Hopf bifurcations and can be numerically continued into parameter regions for which the quasistationary solution is locally linearly stable. We examine how these solutions behave and how they move with respect to different reference frames when the ring is opened and the number of cars goes to infinity.